WA is exactly correct (of course, he works in a gun shop).
There is no "registration" per se. (or, if there is registration, it is both illegal and kept secret).
There are no databases kept AT the BATF (unless you're a conspiracay theorist, as I am, but put that aside for now).
The procedure is this.
When a "gun crime" occurs, the BATF can "trace" (not merely "look up"), but the process is rather burdensome and onerous on the BATF. First of all the gun has to be used in a crime to kick in the procedure. The BATF goes to the manufacturer and asks the manufacturer to tell them from its records, to which distributorship the gun was sold. Then the BATF goes to that distributor, and asks them to provide which FFL the gun was sold to. Then the BATF goes to the actual physical premises of that FFL and manually has to thumb through the 4473s until they find the one with the gun in question (I don't know HOW exactly time-consuming this is, because I don't know what form of organization of the 4473s is used - it may depend on the FFL). Then and only then is the gun linked to a name (and address and.....). From there, there may be more footwork, if the original owner has sold it.
So is that system "registration"? Well, arguably, I suppose, but no, not as registration is defined by most people, and not as registration is defined by the law. The law says the BATF must destroy the information they receive on the NICS checks after 90 days.
We should begin to get worried if the BATF reveals that is knows who owns a particular gun, even though that gun was NOT used in any crime, OR if it knows what gun-crime gun was used in a crime, and never has to show up at the FFL's premises to search the 4473s, to do so.
The hubbub usually comes about when FFL's go out of business, at which time they must by law, turn over all of their 4473s and other documents to the BATF. BATF is still not supposed to compile them into a database (a "registry"), but merely hang onto them in archives for crime-gun searches. If you're like me, with a deep distrust of the gov't, you think they have probably already compiled every nugget of information into the best computer that your tax money can buy, searchable 9 ways to sunday, but are just keeping this fact a secret. On the other hand, if didn't forget to take your meds, or if no prescription for you is necessary, then you believe that they are following the law.
KS, I would be very interested to see that video or a transcript of it, where the head of the BATF is essentially admitting that it breaks the law.